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Algebra Difficulty 3.4 AMC 10/12 Find the answer

Given the function f(x)=sin2x+23sin(x+π4)cos(xπ4)cos2x3f(x)=\sin ^{2}x+2 \sqrt {3}\sin (x+ \frac {π}{4})\cos (x- \frac {π}{4})-\cos ^{2}x- \sqrt {3}.
(1) Find the interval(s) where the function f(x)f(x) is monotonically decreasing.
(2) Find the maximum value of the function f(x)f(x) on the interval [π12,2536π][- \frac {π}{12}, \frac {25}{36}π].

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

(1) The function f(x)=sin2x+23sin(x+π4)cos(xπ4)cos2x3f(x)=\sin ^{2}x+2 \sqrt {3}\sin (x+ \frac {π}{4})\cos (x- \frac {π}{4})-\cos ^{2}x- \sqrt {3} can be simplified as follows:
f(x)=cos2x+23(22sinx+22cosx)(22cosx+22sinx)3f(x)=-\cos 2x+2 \sqrt {3}\cdot( \frac { \sqrt {2}}{2}\sin x+\frac { \sqrt {2}}{2}\cos x)\cdot( \frac { \sqrt {2}}{2}\cos x+\frac { \sqrt {2}}{2}\sin x)- \sqrt {3}
f(x)=cos2x+23(12+12sin2x)3f(x)=-\cos 2x+2 \sqrt {3}\cdot( \frac {1}{2}+\frac {1}{2}\sin 2x)- \sqrt {3}
f(x)=3sin2xcos2x=2sin(2xπ6)f(x)= \sqrt {3}\sin 2x-\cos 2x=2\sin (2x- \frac {π}{6})
Let 2kπ+π22xπ62kπ+3π22kπ+ \frac {π}{2}\leqslant 2x- \frac {π}{6}\leqslant 2kπ+ \frac {3π}{2}, we get kπ+π3xkπ+5π6kπ+ \frac {π}{3}\leqslant x\leqslant kπ +\frac {5π}{6}. Thus, the intervals where the function is decreasing are [kπ+π3,kπ+5π6][kπ+ \frac {π}{3},kπ +\frac {5π}{6}], where kZk\in\mathbb{Z}.

(2) On the interval [π12,2536π][- \frac {π}{12}, \frac {25}{36}π], we have 2xπ6[π3,11π9]2x- \frac {π}{6}\in[- \frac {π}{3}, \frac {11π}{9}]. The maximum value of f(x)f(x) occurs when 2xπ6=π22x- \frac {π}{6}= \frac {π}{2}. Therefore, the maximum value of the function f(x)f(x) is 2\boxed{2}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.